Editing Appendix/Ramblings/ToHadleyAndImamura (section)

==Put It Together==

Figure 2a contains an animation sequence that is intended to illustrate that our empirically constructed radial eigenfunction and accompanying polar-coordinate plots of "constant phase loci"  closely resemble the radial eigenfunctions and constant phase loci that have been displayed in HI11's Figure 16 (a portion of which is redisplayed, here, as Figure 2b).  Each frame of the Figure 2a animation, contains (on the left) a semi-log plot of <math>~f_1(\varpi)</math> versus <math>~\varpi</math> and (on the right) the corresponding constant phase loci that are generated for <math>~\phi_1</math> (top), <math>~\phi_2</math> (middle), and <math>~\phi_3</math> (bottom), when the natural log of the same function, <math>~f_\ln = \ln[f_1(\varpi)]</math>, is plugged into our empirically derived function, <math>~D_{1/2}</math>.  In a [[User:Tohline/Appendix/Ramblings/Azimuthal_Distortions#Specific_Application_to_HI11.27s_Figure_16|separate chapter]] we demonstrate in more detail how each frame of this animation sequence was constructed.  As displayed in Figure 2a, <math>~r_\mathrm{green}</math> is the only parameter that is varied from frame to frame of the animation (the seven specified values are listed in a column to the left of the animation); all of the other empirical model parameters &#8212; <math>~r_\mathrm{blue}</math>, <math>~p</math>, and <math>~\aleph_m</math> &#8212; are held fixed, along with specifications of the inner and outer edges of the torus, <math>~r_-</math> and <math>~r_+</math>.


<div align="center">
<table border="1">
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  <td align="center" colspan="3">
<b><font size="+1">Figure 2:</font></b> &nbsp;
Radial and Azimuthal Eigenfunction Comparison
  </td>
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  <td align="center" colspan="2">'''(a)''' &nbsp; Our Empirically Constructed Function</td>
  <td align="center">'''(b)''' &nbsp; Extracted from Figure 16 of [http://adsabs.harvard.edu/abs/2011Ap%26SS.334....1H HI11]</td>
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<td align="center">

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="center">Specified<p></p><math>~r_\mathrm{green}</math><p></p><hr></td>
  <td align="center">Resulting<p></p><math>~r_\mathrm{min}</math><p></p><hr></td>
</tr>
<tr>
  <td align="center">1.10324</td>
  <td align="center">1.103</td>
</tr>
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  <td align="center">1.05929</td>
  <td align="center">1.077</td>
</tr>
<tr>
  <td align="center">1.01534</td>
  <td align="center">1.059</td>
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<tr>
  <td align="center">0.97139</td>
  <td align="center">1.042</td>
</tr>
<tr>
  <td align="center">0.88349</td>
  <td align="center">1.020</td>
</tr>
<tr>
  <td align="center">0.75164</td>
  <td align="center">0.998</td>
</tr>
<tr>
  <td align="center">0.61979</td>
  <td align="center">0.976</td>
</tr>
</table>

</td>
<td align="center">
[[File:HI11Fig16Animate.gif|350px|Figure 16 from HI11]]
</td>
<td align="center">
[[File:HI11_Fig16ThreeQuarters.png|400px|Figure 16 from HI11]]
</td>
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</table>
</div>

Notice that, by simply varying the value of <math>~r_\mathrm{green}</math>, our semi-log plot of <math>~f_1</math> versus <math>~\varpi</math> sweeps through shapes that qualitatively match the three radial eigenfunctions (see Figure 2b) that arose in the HI11 investigation.  At the largest value, <math>~r_\mathrm{green} = 1.10324</math> (1<sup>st</sup> frame of the animation loop), the minimum of the radial eigenfunction is sharply defined and sits at <math>~r_\mathrm{min} = 1.103</math>, as labeled in the animation; the <math>~f_1(\varpi)</math> eigenfunction displayed in this particular frame closely resembles the radial eigenfunction shown at the top of the HI11 figure and, simultaneously, the <math>~m=1</math> "constant phase locus" plot that appears in the same frame of the animation closely resembles the <math>~m=1</math> constant phase locus that is displayed at the top of the HI11 figure.  When we set <math>~r_\mathrm{green} = 1.05929</math> (2<sup>nd</sup> frame of the animation), the minimum of the radial eigenfunction is still well-defined and sits at <math>~r_\mathrm{min} = 1.077</math>, as labeled in the animation.  The <math>~f_1(\varpi)</math> eigenfunction displayed in this 2<sup>nd</sup> frame closely resembles the radial eigenfunction shown in the middle panel of the HI11 figure and, simultaneously, the <math>~m=2</math> "constant phase locus" plot that appears in this 2<sup>nd</sup> frame of the animation closely resembles the <math>~m=2</math> constant phase locus that is displayed at the middle of the HI11 figure.